Talking Maths in the Bus with a young co-passenger

It was a public holiday – Ganesh Chaturthi. I thought that there wont be much traffic on this day, so I boarded the bus instead of train; but only to repent about this after some time..

Different options were then explored – playing with phone, reading the book, talking to person next to me, but nothing seemed to help that day.  Meanwhile, I found a 11-12 year old boy who I felt was going through the same trouble – killing the time. His mother was sitting 3-4 seats away from him. One could easily perceive his boredom/ uneasiness. I felt the urge to talk to him, but our physical separation was too much to allow this.  Our eyes met sometimes and later we even exchanged smiles but that’s all. However, after few stops, something interesting did happen.  The guy next to me stood up for his destination stop and this boy sprang up and came running to grab this window seat :)  Little did he know that this was a happy-news for me too! But soon I found him so deeply engrossed in watching the life outside the window that I decided to not disturb him (and his learning process) and I resorted to the gadget in my hand. While sifting through the images, an old puzzle caught my attention and I started thinking on improvising it.  And interestingly, the little head too had turned around and his eyes went straight onto my handset. After about 5-7 seconds, when I still found him ‘into’ the puzzle (image), the math-germ within me couldn’t be suppressed more.

“Hi”, I said.

And he looked up at me suddenly with a big & shy smile as if I had caught him doing something red-handed.

“Is it interesting?”

“What is it?”

“A Puzzle! Do you like puzzles?”

“Yes!” again with a big smile J

“Which class you are in?”  (What a silly question!)

“Seventh std.”

I handed-over the phone to him. “Ok! See if you can solve this puzzle then”

Without any hesitation, he grabbed the phone and then asked me for what exactly has to be done.

“What do you see in the image?”

“Cats, dogs and rabbits.”

“Ok. And what else?”

“They are standing on the weighing scale.”

“Ok. So?”

“But what has to be done?”

“You tell me. Anything related to weights?”

“Yes.. Their weights are given.”

 “Are the weights written everywhere?”

“Oh, haaaan, there is a question mark here,..... So do we need to find the weight read by the last scale?”

“What do you feel?”

And he now looks up at me with a confident smile (seems he was finally happy for being able to successfully trace out the question in that puzzle J )  and he further adds “But how to get this?”

“Well, this is what the puzzle is. How to solve this?”  :-)

“I don’t know!”

“Can we try to think from what is given and what is to be found?”

“Weights of animals are given at 3 places. And not given at the last place.”

“Ok.  Are all the places same?”   

“No. Weights of 2 animals are given everywhere. And we need to find the weight of 3 animals.”

“Good. So how will you find?”

“I don’t know.”

“Hmmm.... Let’s look at the first picture”

“Total weight of cat and rabbit is 10kg”

“Ok. So what do you feel what can be their weights?”

“6 and 4.”

“Ok. Who is 6kg and who is 4kg?”

“Cat is 6kg and rabbit is 4kg.”

“Fine. Then what about the dog?”

After 4-5 seconds, “16kg.”

“And how did you get that?”

Pointing at the second picture, “Because rabbit is 4kg, so dog is 16 kg.”

“Ok. Did you look at the 3^rd picture?”

He looked there and after a while, “No...No... Not possible.”

“What is not possible?”

“6 and 4 is not possible.”

“Why?”

“Because then 6+16=22, and it is given that their total is 24.”

“Good. So what do we do then?”

“I don’t know. How to solve this?”

“Hmmm...  From where did you start last time?”

“First picture. 6 and 4.”

“Is anything else possible, if 6 and 4 does not work?”

After a while, “It can be 7 and 3 also”

“Right. But even 4 and 6 could be possible, don’t you feel so?“ 

(Why did I do/ ask this? Because I already knew that 7 and 3 is the right combination and I did not want him to arrive at the right solution so early (easily) ... (Why so? So that I can trouble/tease him more?) I leave this question open for you to think and respond to.)

So he tried for 4 and 6 on his own this time, checking all the 3 images (situations) and explained me with reasoning why this combination is also incorrect.

“So what do we do now?”

“7 and 3”

“Go ahead”  (he had anyways gone ahead w/o waiting for me ;)

“Yes, this is right!”

“What?”

“Cat is 7, rabbit is 3kg”

“Why?”

“Because then dog would be 17kg. And dog and cat would be 17+7=24kg and it is also given as 24kg. It is matching.”

“Hmmm... . Tell me one thing. What if 7 and 3 too would have not worked?”

He thought and replied with confident smile – “We can try (8,2) (9,1) (5,5)”

“Good. So is the puzzle solved?”

After 5-6 seconds and with some finger counting, he replied, “ the answer is 27 kg.”

“How?”

He could explain this quite satisfactorily.

“Do you feel there can be another way to solve this problem?”

After a while, “I don’t know.”

“Think yaaar. You have already solved the problem in one way. So you have the capability to solve it in another way too.”

After seeing him struggle and not responding after 10-15 seconds, I crept in.  “ How many animals are there in each of the figures?”

“Two-two in all and three in the last”

“Ok. What if I bring down the two animals in the 1^st picture and make them stand on the weighing scale of the 3^rd picture?”, I asked him this while moving my hands from top (1^st) to bottom  (3^rd) picture.

“Weight will increase as there will be 4 animals now.”

“Can you describe the new picture clearly to me?”

“We have 1 dog, 2 cats, 1 rabbit and total weight is 34 kg.”

“Good. What do you feel what would I ask you now?”

“Get the animals of 2^nd picture also down?”

“Wow! Can you?”

“So now we have 2 dogs, 2 cats, 2 rabbits and total weight is 54 kg.”

“Good. So?”

No response.

“What do we need to find?”

No response.

“What’s the question?”, while pointing at the last picture.

“Haaaan.... Find the weight of three animals together.”

“Hmmm.... So now?”

No response.

“ What have just found out now?”

“Weight of 2-2 animals is 54kg”

“So?”

“Yes... Yes... So weight of one-one animal will be half of that which is 54/2 = 27kg.”

“Good.  And is your answer right?”

“Yes, I got the same using the 1^st method also.”

“So what do we learn from this.”

“There can be more than 1 method to solve the problem.”

“And can we compare the two methods?”

No response.

“I mean,  what was the approach in both methods, time taken, etc?”

“Yes, the 1st one was very easy, but we have to keep testing all combinations.”

“Do you mean, 2^nd method was difficult?”

“No..Not difficult. But it did not click me so easily.”

“It will.. If you practice few more problems. But is there anything interesting about the 2^nd method?”

“Yes, it’s faster and even smarter.”

‘Hmmm....”  (So happy to hear this!)

“Do you have one more puzzle?” And while saying this he pointed at my handset J

“Oh! You want more?”

“Yes!”

“Ok. Take this. And this is my favorite. I want Rs.100 note from you. But you need to give this to me using 10 notes in all without using Rs10 notes. You can mix the notes if you wish.”

And I instantly found him doing the number crunching. After about 1-2 minutes, he came up with the solution, but it had 11 notes instead of 10 L

“But good. You were very close. I think you will get it soon now.”

And he was again absorbed. What a visual delight it is to watch kids play with numbers J.  Almost in 1-2 minutes, he came up with the correct solution. I can still recollect how satisfied he looked with that sense of achievement.

“ You know there are at least 3 different and correct solutions to this problem.”

“That way, there can be many. It is impossible to find all.”

“Correct. But can we try for some more solutions, at least when we know that 2 more exist?”

And he happily continued for further explorations. After about a minute or so, “ It’s difficult. Can you give me another problem?”

“This itself is another problem for you. 2 more different solutions.”

“I will do that at home. But now Plz give me different problem.”

“Ok.”  And I was about to think and we saw all passengers around us preparing to get down. And then even the boy looked back while his mother signaled him to get down.

“Plz give me the problem fast. We have to get down now.”

“Oh.. I so appreciate your enthusiasm. But let’s get down now. I will give you the next problem when we meet next time and when you give me the solutions for this problem.”

“Arrey,,, but when and how will we meet next time?”

I was almost speechless by now and meanwhile, the mother had arrived by our seat and she intervened, “ I was watching from behind how you kept him so engaged. He was trying and enjoying all the time. Do you take Maths classes?”  ;)

PS: All the above conversations happened in Marathi.